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A fatigue damage parameter for off-axis unidirectional fibre-reinforced composites
International Journal of Fatigue 21 (1999) 849–856 www.elsevier.com/locate/ijfatigue
A fatigue damage parameter for off-axis unidirectional fibrereinforced composites A. Plumtree b
a,*
, G.X. Cheng
b
a Department of Mechanical Engineering, University of Waterloo,Waterloo, ON, Canada, N2L 3G1 Department of Chemical Engineering and Machinery, Xi’an Jiatong University, Xi’an, Shaanxi Province, 710049, People’s Republic of China
Received 11 September 1997; received in revised form 22 February 1998; accepted 28 February 1999
Abstract A fatigue damage parameter based on that of Smith Watson Topper is developed and applied to predict the fatigue-life of offaxis unidirectional fibre reinforced composite materials. A microstress analysis is used to quantify the parameter that takes into account the maximum shear and normal stresses, and shear and normal alternating strains on the fracture plane parallel to the fibres. The major advantage is that this parameter takes into account the effect of fibre orientation and mean stress. By applying this parameter it is possible to develop a set of fatigue lives for various fibre orientation angles and stress ratios using experimental data obtained from different fibre/load angles and stress ratios. The fatigue parameter has been applied to off-axis unidirectional glass/epoxy composite fatigue data. The predicted and experimental results are in good agreement for different fibre/load angles and stress ratios. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue failure parameter; Unidirectional glass/epoxy composite; Mean stress; Fibre/load angle; Smith Watson Topper parameter; Offaxis cyclic loading
1. Introduction One of the concerns in the design of composite components for extended life is the establishment of a general fatigue failure parameter, especially when offaxis loading is considered. Hashin and Rotem [1] presented a simple fatigue failure criterion expressed in terms of S–N curves obtained by uniaxial cyclic testing of off-axis unidirectional specimens. This criterion allowed plane stress fatigue failure to be predicted with reasonable accuracy. Later Awerbuch and Hahn [2] performed off-axis fatigue tests on composite laminae and, after fitting their experimental data using a power law equation, concluded that a more detailed investigation was necessary. Talreja ([3] pp. 137-53) undertook to clarify the complex nature of the effects of fibre properties on composite fatigue performance by introducing a fatigue damage criterion based on conceptual fatigue-life
* Corresponding author. Tel.: +1-519-888-4567; fax: +1-519-8886197. E-mail address: [email protected] (A. Plumtree)
diagrams requiring information on rate-controlling parameters. Reifsnider and Gao [4] presented a micromechanics model for the prediction of fatigue-life, which considered the interaction between fibres and matrix, as well as interfacial bonding. Based on damage examination, Kellas et al. [5] suggested a parameter defined as the damage transition stress which, unfortunately, was difficult to determine. Although these and other studies have been carried out, there is still a need for further research concerning the establishment of a general fatigue damage parameter for composite life prediction. Several reviews have been conducted on cumulative fatigue damage and life theories (e.g. [6]). In general, they can be divided into three main groups, namely: equivalent stress or strain, critical plane and energybased [7]. The most popular stress-based criteria are extensions of classical maximum principal stress and maximum shear stress failure theories. Normally, the maximum principal stress theory is used when fatigue is dominated by tensile stresses whereas the Von Mises and Tresca theories are applied to shear stress dominated fatigue. Strain-based approaches have given satisfactory results for uniaxial low-cycle fatigue. However, it is
0142-1123/99/$ - see front matter. 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 9 9 ) 0 0 0 2 6 - 2
850
A. Plumtree, G.X. Cheng / International Journal of Fatigue 21 (1999) 849–856
often found that even strain-based criteria are insufficient to encompass the wide variety of loading modes encountered. Stress-or strain-based criteria do not express the interaction between stress and strain during deformation. The critical plane concept was proposed by Findley [8] and Miller and Brown [9]. Good correlation with multiaxial fatigue data was obtained for isotropic metals [10]. The major advantages of this approach are its ability to predict the plane on which cracks occur and to facilitate the construction of fatigue-life plots. Based on this concept, Glinka et al. [11] proposed a multiaxial fatigue parameter which considered mean stress effects by combining the maximum shear and tensile stresses and alternating shear stresses and strains on the critical plane. The validity of their fatigue parameter was verified using isotropic metals. By contrast, unidirectional fibre reinforced composites are orthotropic. Multi-damage mechanisms and fibre-matrix interaction must be considered. Shen et al. [12] introduced the critical plane concept for composite life prediction by relating the local strains on the fracture plane to fatigue-life. In the case of off-axially loaded unidirectional composites, fracture occurred in the matrix parallel to the fibre axis. Their fatigue control parameter accounted for the shear and normal strain ranges on this plane, and correlated the fatigue lives of composites tested under a variety of multiaxial loading conditions and geometrical configurations. However, mean stress effects were not taken into consideration. Energy-based criteria have the advantage that they incorporate both stress and strain, and have been applied to correlate experimental fatigue data by analyzing stress–strain hysteresis loops. Garud [13] related the plastic work done per cycle to fatigue-life. Both axial and torsional stresses as well as plastic axial and torsional strain increments were taken into account. For better correlation with tension–torsion data, the torsional work done was weighted half that of the axial work done. Other weighting factors have since been suggested [7]. Kadi and Ellyin [14] proposed a fatigue failure criterion for unidirectional composite laminae based on input energy which could also allow for mean stress effects. Assuming that the stress–strain relationship was essentially elastic and that the maximum and minimum stresses were positive, the cyclic strain energy, equivalent to the area under the stress–strain curve, would be expressed. By dividing the strain energy by the maximum monotonic strain energy, the effect of fibre orientation angle was indirectly taken into account. In general, the fatigue-life of unidirectional composites decreases with increase in fibre/load angle and mean stress for a given stress amplitude. In the absence of a well defined characteristic method or parameter that can be used to predict the fatigue performance of unidirectional composite materials, extensive tests are carried out for different fibre orientations and loading con-
ditions. In an attempt to reduce this lengthy and time consuming procedure, the present work was undertaken to investigate the feasibility of such a fatigue parameter and establish a life prediction model that takes into account fibre orientation angle and mean stress. The approach adopted may be described as a macro-concept using micro-mechanics. The Smith Watson Topper (SWT) parameter [15] is frequently used to describe uniaxial fatigue mean stress effects. Recent work has demonstrated that it can be used to correlate crack initiation and growth data for metals subjected to multiaxial fatigue [16]. For small plastic strains, the SWT parameter allows for comparison of different mean normal stresses by converting them to an equivalent zero mean stress range (⌬s0) for a particular life: (⌬s0)2⫽smax⌬s
(1)
where smax⫽sm⫹⌬s/2 and sm is the mean normal stress, and ⌬s/2 is the normal stress amplitude. The advantage of the equivalent zero mean stress concept is that a material constant is not required. It has been found particularly effective when considering the fatigue data of aluminum alloys and steels [15]. In its original form the SWT parameter was written: PN⫽smax⌬e
(2)
where ⌬e is the tensile strain range and PN, the mean normal stress fatigue parameter, is constant for a given life. This parameter has the same units as strain energy density. In multiaxial cases the shear stress parameter (Ps) is taken into account by adding the two parameters: PN⫹Ps⫽smax⌬e⫹tmax⌬g
(3)
where tmax is the maximum shear stress and ⌬g is the shear strain range. Eq. (3) will be developed for off-axis unidirectional composites.
2. Fatigue damage parameter Fig. 1 is a schematic indicating the normal and shear stresses on the plane parallel to the fibres. In this case the composite is loaded at an angle q to the fibres. The strength is dominated by the matrix properties. Once a crack has formed in the matrix, its tip will be subjected to two displacements—an opening mode normal to the fibres (s22) and in-plane sliding or shear parallel to the fibres (t12). This will lead to mixed-mode crack growth parallel to the fibres. For an elemental cube subjected to tensile cyclic stress, the local normal parameter, accounting for tensile mean stresses [Eq. (2)] is expressed by:
A. Plumtree, G.X. Cheng / International Journal of Fatigue 21 (1999) 849–856
Fig. 1. Coordinate system and schematic of stress state at fibre– matrix interface.
⌬W1*⫽smax 22 ⌬e22
(4)
For plane stress or strain, the shear deformation at the fibre/matrix interface is ⌬g12/2 and the local parameter for an element at the interface subjected to shear is: ⌬W*2⫽tmax 12 ⌬g12/2
851
is capable of predicting stress–strain response for any combination of these loading modes. The fibre and matrix properties given in Table 1 were used to determine the normal and shear stresses and strains on the fracture plane parallel to the fibres. In the constant strain triangle finite element method used in the deformation analysis to calculate the microstrain fields in the fibre and matrix, the interfacial region was assigned the same properties as the matrix. Because of symmetry, a quadrant of the fibre and surrounding matrix was analyzed, involving 207 elements and 124 nodes. The fatigue parameter, ⌬W*, was determined for the applied test conditions corresponding to a particular life when matrix failure occurred. Crack growth parallel to the fibres takes place because of the high degree of orthotropy in the strength of the unidirectional composite. This was found to be the case for the experimental fatigue data [1] examined. Unidirectional E-glass fibre reinforcements in epoxy were tested under cyclic axial loading at various fibre/load orientations. All specimens failed in the matrix where the cracks initiated and grew along planes parallel to the fibres.
(5)
Combination of the normal and shear components on this plane is expressed by:
3. Verification
max ⌬W*⫽⌬W*1⫹⌬W*2⫽smax 22 ⌬e22⫹t12 ⌬g12/2
3.1. Fatigue-life predictions for different fibre orientations
(6)
The fatigue parameter, ⌬W*, will be constant for a given fatigue-life and the mean stresses are taken into account since the maximum normal stress smax 22 is defined by: m smax 22 ⫽s22⫹⌬s22/2
(7) max 12
and the maximum shear stress t is chosen from the greater of the two extreme values of the shear stress, i.e. m m tmax 12 ⫽兩t12⫹⌬t12/2兩 or 兩t12⫺⌬t12/2兩
(8)
Consequently, the effect of stress ratio on fatigue behaviour is considered by using the parameter ⌬W*. It also gives the direction of the fracture plane. 2.1. Calculation of microstresses on the fracture plane A microstress analysis has been carried out on E-glass fibre reinforced epoxy composites using numerical modelling. This procedure has been described by Shen et al. [12] and proved to be successful despite limited FEM data for cyclically loaded unidirectional composites. The form of the generalized plane strain analysis assumes that displacements can occur in all three coordinate directions. Each displacement is dependent upon the 2 and 3 coordinates, and the 1-direction displacement (fibre direction) has an additional linear dependence. Therefore, although the analysis is basically two-dimensional in nature, five stress components (s11, s22, s33, t12 and t13) can be modelled. The finite element program
Hashin and Rotem [1] tested unidirectional specimens containing 0.6 volume fraction E-glass fibres cycled at a stress ratio R(=minimum stress/maximum stress) of 0.1 with fibre/load axis angles, q, of 5°, 10°, 15°, 20°, 30° and 60°. As expected, the original maximum stress vs log cycle curves indicated that the fatigue lives were strongly dependent upon the fibre/load angle. The fatigue parameter, ⌬W*, was applied to correlate these experimental data, and was plotted against the number of reversals to failure, 2Nf, using a log–log coordinate system, shown in Fig. 2. Within experimental error, all the data collapsed onto a straight line expressed by, lg(⌬W*)⫽0.06⫺0.13 lg(2Nf)
(9)
This equation covers a wide range of fibre/load angles
Table 1 Properties of fibre and matrix Materials
Elastic modulus E (MPa)
Poisson’s ratio n
Tensile strength (MPa)
E-glass (rf=5.0 µm) Epoxy
7.24×104
0.22
1726
0.39×104
0.37
38.7
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A. Plumtree, G.X. Cheng / International Journal of Fatigue 21 (1999) 849–856
Fig. 2.
Correlation between the unified parameter (⌬W*) and number of reversals (2Nf) for various fibre-load orientations (R=0.1).
that includes both shear and transverse failure mechanisms. In general terms Eq. (9) may be written, lg(⌬W*)⫽αlg(2Nf)⫹lgβ
(10)
where α is the slope of the log ⌬W*–log 2Nf plot. Both α and β are material constants. The unified fatigue plot (log ⌬W* vs log 2Nf) created from experimental data is applicable to 5–60° fibre/load angles for a given R ratio. Conversely, using the data for one fibre/load angle it should be possible to predict the fatigue-life of the same composite tested at a different fibre/load angle and same R ratio. To verify this approach, the fatigue data for a fibre/load angle of 15° was analyzed and the best fit relationship determined by the least squares method was found to be: lg(⌬W*)⫽0.20⫺0.14 lg(2Nf)
(11)
Eq. (11) was then used to predict the fatigue lives of specimens tested at the other angles of 5°, 10°, 20°, 30° and 60°. A comparison of the predicted lives given by Eq. (11) and the original experimental results is shown in Fig. 3. Although shear failure occurs at low fibre/load angles and transverse failure at high angles, good agreement between the predicted results and experimental data for all fibre/load angles is apparent.
3.2. Fatigue-life prediction for different mean stresses Considering the influence of mean stress on the fatigue-life of metals, it must be pointed out that excellent models have already been established. Two of the most commonly used for metals (isotropic cases) are the Morrow mean stress model and the SWT parameter, mentioned earlier. The latter has been extended to orthotropic materials in this present work. In order to verify the capability of the fatigue parameter ⌬W* to correlate fatigue data under different mean stresses, a second set of data [14] was analyzed. Epoxy specimens reinforced with 0.5 volume fraction unidirectional E-glass fibres were cycled by Kadi and Ellyin [14] at three stress ratios, R=⫺1, R=0 and R=0.5, as well as different fibre/load angles, q, of 19°, 45° and 71°. The influence of stress ratio on fatigue-life is represented in Fig. 4 which gives the log–log plot of maximum stress, Smax, vs number of reversals to failure, 2Nf, for q=19°. Fatigue life decreased with decrease in stress ratio for a given maximum stress. Plots for the other fibre/load angles (q=45° and 71°) showed the same trend. As expected, for a constant stress ratio and given life, the Smax values for q=19° were the highest, whereas those for q=71° were lowest. By applying ⌬W* the data for the various stress ratios shifted onto a common curve,
A. Plumtree, G.X. Cheng / International Journal of Fatigue 21 (1999) 849–856
Fig. 3.
853
Comparison of predicted fatigue life using q=15° with experimental data for other angles (R=0.1).
shown in Fig. 5. The fatigue lives of the unidirectional composite tested at different fibre orientation angles for the three stress ratios were expressed by the following empirical relationship, lg⌬W*⫽0.27⫺0.11 lg(2Nf)
(12)
The relatively large amount of scatter seen in Fig. 5 may be the result of including the R=⫺1 data which are influenced by fibre buckling in addition to matrix cracking. Using this approach, it is possible to estimate the fatigue lives of laminae for different fibre/load angles and stress ratios using other loading conditions, especially those with positive mean stresses. Considering the data obtained from those specimens tested at a stress ratio of R=0 and a fibre/load angle of 45°, the relationship between ⌬W* and 2Nf was found to be: lg(⌬W*)⫽0.46⫺0.14 lg(2Nf)
(13)
This equation was then applied to describe the fatigue lives of specimens tested at other fibre/load angles and stress ratios. Fig. 6 shows the predicted Smax vs 2Nf curve for R=0.5 and q=71°. Numerical results generated using this fatigue parameter compare well with the experimental values, illustrating its general applicability.
4. Discussion The fatigue parameter ⌬W* is capable of predicting fatigue lives of off-axis composites tested at different fibre/load angles and mean stresses with reasonable accuracy. Its versatility has been demonstrated in Figs. 3 and 6. In the former figure using a fibre/load angle of 15°, ⌬W* was determined from specimens whose lives were dominated by shear failure. This information was then applied to other fibre/load angles including 60° where transverse failure occurred. Hence, ⌬W* applies to both failure mechanisms. Considering the general power law expression used [Eq. (10)], which was based on the limited data available, it is apparent that the material constant α is relatively independent (0.11⬍α⬍0.14) of fibre/load angle, mean stress, fibre content and type of epoxy. The values of β show more scatter (1.2–1.6 for 0.6 fibre volume fraction and 1.9–2.9 for 0.5 fibre volume fraction composites). Further investigations will be conducted on larger sample populations to establish the extent of variability of these constants. However, the present approach illustrates that a general fatigue damage parameter accrues from analysis of the local stresses and strains in the matrix on the fracture
854
A. Plumtree, G.X. Cheng / International Journal of Fatigue 21 (1999) 849–856
Fig. 4.
Maximum stress plotted against number of reversals to failure for three stress ratios (q=19°).
plane in off-axis unidirectional composites. It holds promise for analysing multiaxial fatigue data. In fact, Chu et al. [16] have shown that the SWT parameter, in a different form, satisfactorily models the stress–strain behaviour and fatigue-life prediction of steel shafts subjected to multiaxial loading. In their case, since the material was isotropic, only the macrostresses and strains were considered. Typical of other interactive failure models for orthotropic unidirectional composites the present fatigue parameter does not distinguish between failure mechanisms for a given fibre/load angle. However, since matrix failure is dominant, comparison may be made with simple interactive models for static loading such as the Pucksimple failure criterion [17] derived on the assumption that matrix failures are determined by s22 and t12, and fibre failures are determined by s11:
冉 冊 冉冊
s22 2 t12 2 ⫹ ⫽1 Y Q
(14)
where s22 and Y are the respective tensile and failure stresses normal to the fibre (in-plane), and t12 and Q are the respective shear and failure stresses in the 1–2 plane. Eq. (14) can be rearranged assuming linear elastic behaviour
s22e22⫹t12g12K⫽K⬘
(15)
where ⑀22 and E22 are the strain and elastic modulus normal to the fibre (in-plane), g12 and G12 are the shear strain and modulus in the 1–2 plane, K=G12 Y2/E22 Q2 and K⬘=Y2/E22. Eqs. (6) and (15) have the same form and respectively express the combination of the tensile and shear stresses and strains required to initiate failure under static and cyclic loading conditions in the matrix of an off-axis unidirectional composite. 5. Conclusions A parameter has been proposed to predict the fatigue lives of unidirectional composites loaded off-axis. The fatigue parameter includes the maximum shear and normal stresses as well as the normal and shear stress ranges on the fracture plane parallel to the fibre axis. It is load path dependent. This approach could possibly reduce the number of fatigue tests required to characterize a particular fibre/load orientation or mean stress for an off-axis unidirectional composite. The fatigue parameter has been applied to predict the lives of unidirectional E-glass epoxy reinforced composites cycled at different fibre/load angles and mean
A. Plumtree, G.X. Cheng / International Journal of Fatigue 21 (1999) 849–856
Fig. 5.
855
Unified fatigue parameter (⌬W*) plotted against number of reversals to failure (2Nf) for different stress ratios.
conditions it holds promise for application to those cases where unidirectional composites are used for components that are cycled under multiaxial loading conditions.
Acknowledgements
Fig. 6. Predicted curve for Smax versus 2Nf for R=0.5 and q=71° data using R=0 and q=45° data.
stresses. The predicted data are in good agreement with the experimental results. It is shown that by applying the fatigue parameter derived from experimental data for one fibre/load angle, the fatigue lives of the same composite with other fibre/load angles may be predicted. A similar approach may be applied for mean stresses. Since the fatigue parameter appears capable of correlating fatigue data obtained under a variety of loading
This study was supported by the National Science and Engineering Research Council of Canada (NSERC) under Grant 0GP 0002770 and National Nature Science Foundation of China (NNSFC). The authors wish to thank Dr G. Wang, Dr L. Shi, Mr Y. Huang and Mr J. Petermann for valuable discussions and Marlene Dolson for typing the manuscript.
References [1] Hashin Z, Rotem R. A fatigue failure criterion for fibre reinforced materials. J Comp Mater 1973;7:448–64. [2] Awerbuch J, Hahn HT. Off-axis fatigue of graphite/epoxy composites, fatigue of fibrous, composite materials. ASTM STP 723, 1981;243–73. [3] Talreja R. Damage models for fatigue of composite materials, fatigue and creep of composite materials. Denmark: Riso National Laboratory, 1982.
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[4] Reifsnider KL, Gao Z. A micromechanics model for composites under fatigue loading. Int J Fat 1991;13(2):149–56. [5] Kellas S, Morton J, Curtis PT. A characteristic fatigue parameter for notched composite. Int J Fat 1991;13(1):35–43. [6] Fatemi A, Yang L. Cumulative fatigue damage and life prediction theories, a survey of the state of the art for homogeneous materials. Int J Fat 1998;20:9–34. [7] Bannantine JA. A review of multiaxial fatigue damage model and observed material behaviour. In: Muralidharan U, editor. Impact of improved material quality on properties, product performance and design. ASME, 1991:213–37. [8] Findley WN. A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending. J Engng Ind Trans ASME 1959;18:301–6. [9] Brown MW, Miller KJ. A theory for fatigue under multiaxial stress–strain conditions. Proc Inst Mech Engng 1973;187:745–55. [10] McDiarmid DL. A general criterion of high cycle multiaxial fatigue failure. J Fat Fract Engng Mater Struct 1991;14(4):429– 53.
[11] Glinka G, Wang G, Plumtree A. Mean stress effects in multiaxial fatigue. J Fat Fract Engng Mater Struct 1995;18(7-8):755–64. [12] Shen G, Glinka G, Plumtree A. Fatigue life prediction of notched composite component. J Fat Fract Engng Mater Struct 1994;17(1):77–91. [13] Garud YS. Multiaxial fatigue: a survey of the state of the art. J Test Eval 1981;9:165–78. [14] El Kadi H, Ellyin F. Effect of stress ratio on the fatigue of unidirectional glass fibre/epoxy composite laminae. Composites 1994;10:917–24. [15] Smith KN, Watson P, Topper TH. A stress strain function for the fatigue of metals. J Mater, JMLSA 1970;5:767–78. [16] Chu C-C, Conle FA, Bonnen JJF. Multiaxial stress–strain modelling and life prediction of SAE axle shafts. In: McDowell DL, Ellis R, editors. Advances in multiaxial fatigue ASTM STP 1191. ASTM, 1993:37–54. [17] Puck A. Kunstoffe 1969;59:780.
A fatigue damage parameter for off-axis unidirectional fibrereinforced composites A. Plumtree b
a,*
, G.X. Cheng
b
a Department of Mechanical Engineering, University of Waterloo,Waterloo, ON, Canada, N2L 3G1 Department of Chemical Engineering and Machinery, Xi’an Jiatong University, Xi’an, Shaanxi Province, 710049, People’s Republic of China
Received 11 September 1997; received in revised form 22 February 1998; accepted 28 February 1999
Abstract A fatigue damage parameter based on that of Smith Watson Topper is developed and applied to predict the fatigue-life of offaxis unidirectional fibre reinforced composite materials. A microstress analysis is used to quantify the parameter that takes into account the maximum shear and normal stresses, and shear and normal alternating strains on the fracture plane parallel to the fibres. The major advantage is that this parameter takes into account the effect of fibre orientation and mean stress. By applying this parameter it is possible to develop a set of fatigue lives for various fibre orientation angles and stress ratios using experimental data obtained from different fibre/load angles and stress ratios. The fatigue parameter has been applied to off-axis unidirectional glass/epoxy composite fatigue data. The predicted and experimental results are in good agreement for different fibre/load angles and stress ratios. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue failure parameter; Unidirectional glass/epoxy composite; Mean stress; Fibre/load angle; Smith Watson Topper parameter; Offaxis cyclic loading
1. Introduction One of the concerns in the design of composite components for extended life is the establishment of a general fatigue failure parameter, especially when offaxis loading is considered. Hashin and Rotem [1] presented a simple fatigue failure criterion expressed in terms of S–N curves obtained by uniaxial cyclic testing of off-axis unidirectional specimens. This criterion allowed plane stress fatigue failure to be predicted with reasonable accuracy. Later Awerbuch and Hahn [2] performed off-axis fatigue tests on composite laminae and, after fitting their experimental data using a power law equation, concluded that a more detailed investigation was necessary. Talreja ([3] pp. 137-53) undertook to clarify the complex nature of the effects of fibre properties on composite fatigue performance by introducing a fatigue damage criterion based on conceptual fatigue-life
* Corresponding author. Tel.: +1-519-888-4567; fax: +1-519-8886197. E-mail address: [email protected] (A. Plumtree)
diagrams requiring information on rate-controlling parameters. Reifsnider and Gao [4] presented a micromechanics model for the prediction of fatigue-life, which considered the interaction between fibres and matrix, as well as interfacial bonding. Based on damage examination, Kellas et al. [5] suggested a parameter defined as the damage transition stress which, unfortunately, was difficult to determine. Although these and other studies have been carried out, there is still a need for further research concerning the establishment of a general fatigue damage parameter for composite life prediction. Several reviews have been conducted on cumulative fatigue damage and life theories (e.g. [6]). In general, they can be divided into three main groups, namely: equivalent stress or strain, critical plane and energybased [7]. The most popular stress-based criteria are extensions of classical maximum principal stress and maximum shear stress failure theories. Normally, the maximum principal stress theory is used when fatigue is dominated by tensile stresses whereas the Von Mises and Tresca theories are applied to shear stress dominated fatigue. Strain-based approaches have given satisfactory results for uniaxial low-cycle fatigue. However, it is
0142-1123/99/$ - see front matter. 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 9 9 ) 0 0 0 2 6 - 2
850
A. Plumtree, G.X. Cheng / International Journal of Fatigue 21 (1999) 849–856
often found that even strain-based criteria are insufficient to encompass the wide variety of loading modes encountered. Stress-or strain-based criteria do not express the interaction between stress and strain during deformation. The critical plane concept was proposed by Findley [8] and Miller and Brown [9]. Good correlation with multiaxial fatigue data was obtained for isotropic metals [10]. The major advantages of this approach are its ability to predict the plane on which cracks occur and to facilitate the construction of fatigue-life plots. Based on this concept, Glinka et al. [11] proposed a multiaxial fatigue parameter which considered mean stress effects by combining the maximum shear and tensile stresses and alternating shear stresses and strains on the critical plane. The validity of their fatigue parameter was verified using isotropic metals. By contrast, unidirectional fibre reinforced composites are orthotropic. Multi-damage mechanisms and fibre-matrix interaction must be considered. Shen et al. [12] introduced the critical plane concept for composite life prediction by relating the local strains on the fracture plane to fatigue-life. In the case of off-axially loaded unidirectional composites, fracture occurred in the matrix parallel to the fibre axis. Their fatigue control parameter accounted for the shear and normal strain ranges on this plane, and correlated the fatigue lives of composites tested under a variety of multiaxial loading conditions and geometrical configurations. However, mean stress effects were not taken into consideration. Energy-based criteria have the advantage that they incorporate both stress and strain, and have been applied to correlate experimental fatigue data by analyzing stress–strain hysteresis loops. Garud [13] related the plastic work done per cycle to fatigue-life. Both axial and torsional stresses as well as plastic axial and torsional strain increments were taken into account. For better correlation with tension–torsion data, the torsional work done was weighted half that of the axial work done. Other weighting factors have since been suggested [7]. Kadi and Ellyin [14] proposed a fatigue failure criterion for unidirectional composite laminae based on input energy which could also allow for mean stress effects. Assuming that the stress–strain relationship was essentially elastic and that the maximum and minimum stresses were positive, the cyclic strain energy, equivalent to the area under the stress–strain curve, would be expressed. By dividing the strain energy by the maximum monotonic strain energy, the effect of fibre orientation angle was indirectly taken into account. In general, the fatigue-life of unidirectional composites decreases with increase in fibre/load angle and mean stress for a given stress amplitude. In the absence of a well defined characteristic method or parameter that can be used to predict the fatigue performance of unidirectional composite materials, extensive tests are carried out for different fibre orientations and loading con-
ditions. In an attempt to reduce this lengthy and time consuming procedure, the present work was undertaken to investigate the feasibility of such a fatigue parameter and establish a life prediction model that takes into account fibre orientation angle and mean stress. The approach adopted may be described as a macro-concept using micro-mechanics. The Smith Watson Topper (SWT) parameter [15] is frequently used to describe uniaxial fatigue mean stress effects. Recent work has demonstrated that it can be used to correlate crack initiation and growth data for metals subjected to multiaxial fatigue [16]. For small plastic strains, the SWT parameter allows for comparison of different mean normal stresses by converting them to an equivalent zero mean stress range (⌬s0) for a particular life: (⌬s0)2⫽smax⌬s
(1)
where smax⫽sm⫹⌬s/2 and sm is the mean normal stress, and ⌬s/2 is the normal stress amplitude. The advantage of the equivalent zero mean stress concept is that a material constant is not required. It has been found particularly effective when considering the fatigue data of aluminum alloys and steels [15]. In its original form the SWT parameter was written: PN⫽smax⌬e
(2)
where ⌬e is the tensile strain range and PN, the mean normal stress fatigue parameter, is constant for a given life. This parameter has the same units as strain energy density. In multiaxial cases the shear stress parameter (Ps) is taken into account by adding the two parameters: PN⫹Ps⫽smax⌬e⫹tmax⌬g
(3)
where tmax is the maximum shear stress and ⌬g is the shear strain range. Eq. (3) will be developed for off-axis unidirectional composites.
2. Fatigue damage parameter Fig. 1 is a schematic indicating the normal and shear stresses on the plane parallel to the fibres. In this case the composite is loaded at an angle q to the fibres. The strength is dominated by the matrix properties. Once a crack has formed in the matrix, its tip will be subjected to two displacements—an opening mode normal to the fibres (s22) and in-plane sliding or shear parallel to the fibres (t12). This will lead to mixed-mode crack growth parallel to the fibres. For an elemental cube subjected to tensile cyclic stress, the local normal parameter, accounting for tensile mean stresses [Eq. (2)] is expressed by:
A. Plumtree, G.X. Cheng / International Journal of Fatigue 21 (1999) 849–856
Fig. 1. Coordinate system and schematic of stress state at fibre– matrix interface.
⌬W1*⫽smax 22 ⌬e22
(4)
For plane stress or strain, the shear deformation at the fibre/matrix interface is ⌬g12/2 and the local parameter for an element at the interface subjected to shear is: ⌬W*2⫽tmax 12 ⌬g12/2
851
is capable of predicting stress–strain response for any combination of these loading modes. The fibre and matrix properties given in Table 1 were used to determine the normal and shear stresses and strains on the fracture plane parallel to the fibres. In the constant strain triangle finite element method used in the deformation analysis to calculate the microstrain fields in the fibre and matrix, the interfacial region was assigned the same properties as the matrix. Because of symmetry, a quadrant of the fibre and surrounding matrix was analyzed, involving 207 elements and 124 nodes. The fatigue parameter, ⌬W*, was determined for the applied test conditions corresponding to a particular life when matrix failure occurred. Crack growth parallel to the fibres takes place because of the high degree of orthotropy in the strength of the unidirectional composite. This was found to be the case for the experimental fatigue data [1] examined. Unidirectional E-glass fibre reinforcements in epoxy were tested under cyclic axial loading at various fibre/load orientations. All specimens failed in the matrix where the cracks initiated and grew along planes parallel to the fibres.
(5)
Combination of the normal and shear components on this plane is expressed by:
3. Verification
max ⌬W*⫽⌬W*1⫹⌬W*2⫽smax 22 ⌬e22⫹t12 ⌬g12/2
3.1. Fatigue-life predictions for different fibre orientations
(6)
The fatigue parameter, ⌬W*, will be constant for a given fatigue-life and the mean stresses are taken into account since the maximum normal stress smax 22 is defined by: m smax 22 ⫽s22⫹⌬s22/2
(7) max 12
and the maximum shear stress t is chosen from the greater of the two extreme values of the shear stress, i.e. m m tmax 12 ⫽兩t12⫹⌬t12/2兩 or 兩t12⫺⌬t12/2兩
(8)
Consequently, the effect of stress ratio on fatigue behaviour is considered by using the parameter ⌬W*. It also gives the direction of the fracture plane. 2.1. Calculation of microstresses on the fracture plane A microstress analysis has been carried out on E-glass fibre reinforced epoxy composites using numerical modelling. This procedure has been described by Shen et al. [12] and proved to be successful despite limited FEM data for cyclically loaded unidirectional composites. The form of the generalized plane strain analysis assumes that displacements can occur in all three coordinate directions. Each displacement is dependent upon the 2 and 3 coordinates, and the 1-direction displacement (fibre direction) has an additional linear dependence. Therefore, although the analysis is basically two-dimensional in nature, five stress components (s11, s22, s33, t12 and t13) can be modelled. The finite element program
Hashin and Rotem [1] tested unidirectional specimens containing 0.6 volume fraction E-glass fibres cycled at a stress ratio R(=minimum stress/maximum stress) of 0.1 with fibre/load axis angles, q, of 5°, 10°, 15°, 20°, 30° and 60°. As expected, the original maximum stress vs log cycle curves indicated that the fatigue lives were strongly dependent upon the fibre/load angle. The fatigue parameter, ⌬W*, was applied to correlate these experimental data, and was plotted against the number of reversals to failure, 2Nf, using a log–log coordinate system, shown in Fig. 2. Within experimental error, all the data collapsed onto a straight line expressed by, lg(⌬W*)⫽0.06⫺0.13 lg(2Nf)
(9)
This equation covers a wide range of fibre/load angles
Table 1 Properties of fibre and matrix Materials
Elastic modulus E (MPa)
Poisson’s ratio n
Tensile strength (MPa)
E-glass (rf=5.0 µm) Epoxy
7.24×104
0.22
1726
0.39×104
0.37
38.7
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Fig. 2.
Correlation between the unified parameter (⌬W*) and number of reversals (2Nf) for various fibre-load orientations (R=0.1).
that includes both shear and transverse failure mechanisms. In general terms Eq. (9) may be written, lg(⌬W*)⫽αlg(2Nf)⫹lgβ
(10)
where α is the slope of the log ⌬W*–log 2Nf plot. Both α and β are material constants. The unified fatigue plot (log ⌬W* vs log 2Nf) created from experimental data is applicable to 5–60° fibre/load angles for a given R ratio. Conversely, using the data for one fibre/load angle it should be possible to predict the fatigue-life of the same composite tested at a different fibre/load angle and same R ratio. To verify this approach, the fatigue data for a fibre/load angle of 15° was analyzed and the best fit relationship determined by the least squares method was found to be: lg(⌬W*)⫽0.20⫺0.14 lg(2Nf)
(11)
Eq. (11) was then used to predict the fatigue lives of specimens tested at the other angles of 5°, 10°, 20°, 30° and 60°. A comparison of the predicted lives given by Eq. (11) and the original experimental results is shown in Fig. 3. Although shear failure occurs at low fibre/load angles and transverse failure at high angles, good agreement between the predicted results and experimental data for all fibre/load angles is apparent.
3.2. Fatigue-life prediction for different mean stresses Considering the influence of mean stress on the fatigue-life of metals, it must be pointed out that excellent models have already been established. Two of the most commonly used for metals (isotropic cases) are the Morrow mean stress model and the SWT parameter, mentioned earlier. The latter has been extended to orthotropic materials in this present work. In order to verify the capability of the fatigue parameter ⌬W* to correlate fatigue data under different mean stresses, a second set of data [14] was analyzed. Epoxy specimens reinforced with 0.5 volume fraction unidirectional E-glass fibres were cycled by Kadi and Ellyin [14] at three stress ratios, R=⫺1, R=0 and R=0.5, as well as different fibre/load angles, q, of 19°, 45° and 71°. The influence of stress ratio on fatigue-life is represented in Fig. 4 which gives the log–log plot of maximum stress, Smax, vs number of reversals to failure, 2Nf, for q=19°. Fatigue life decreased with decrease in stress ratio for a given maximum stress. Plots for the other fibre/load angles (q=45° and 71°) showed the same trend. As expected, for a constant stress ratio and given life, the Smax values for q=19° were the highest, whereas those for q=71° were lowest. By applying ⌬W* the data for the various stress ratios shifted onto a common curve,
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Fig. 3.
853
Comparison of predicted fatigue life using q=15° with experimental data for other angles (R=0.1).
shown in Fig. 5. The fatigue lives of the unidirectional composite tested at different fibre orientation angles for the three stress ratios were expressed by the following empirical relationship, lg⌬W*⫽0.27⫺0.11 lg(2Nf)
(12)
The relatively large amount of scatter seen in Fig. 5 may be the result of including the R=⫺1 data which are influenced by fibre buckling in addition to matrix cracking. Using this approach, it is possible to estimate the fatigue lives of laminae for different fibre/load angles and stress ratios using other loading conditions, especially those with positive mean stresses. Considering the data obtained from those specimens tested at a stress ratio of R=0 and a fibre/load angle of 45°, the relationship between ⌬W* and 2Nf was found to be: lg(⌬W*)⫽0.46⫺0.14 lg(2Nf)
(13)
This equation was then applied to describe the fatigue lives of specimens tested at other fibre/load angles and stress ratios. Fig. 6 shows the predicted Smax vs 2Nf curve for R=0.5 and q=71°. Numerical results generated using this fatigue parameter compare well with the experimental values, illustrating its general applicability.
4. Discussion The fatigue parameter ⌬W* is capable of predicting fatigue lives of off-axis composites tested at different fibre/load angles and mean stresses with reasonable accuracy. Its versatility has been demonstrated in Figs. 3 and 6. In the former figure using a fibre/load angle of 15°, ⌬W* was determined from specimens whose lives were dominated by shear failure. This information was then applied to other fibre/load angles including 60° where transverse failure occurred. Hence, ⌬W* applies to both failure mechanisms. Considering the general power law expression used [Eq. (10)], which was based on the limited data available, it is apparent that the material constant α is relatively independent (0.11⬍α⬍0.14) of fibre/load angle, mean stress, fibre content and type of epoxy. The values of β show more scatter (1.2–1.6 for 0.6 fibre volume fraction and 1.9–2.9 for 0.5 fibre volume fraction composites). Further investigations will be conducted on larger sample populations to establish the extent of variability of these constants. However, the present approach illustrates that a general fatigue damage parameter accrues from analysis of the local stresses and strains in the matrix on the fracture
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Fig. 4.
Maximum stress plotted against number of reversals to failure for three stress ratios (q=19°).
plane in off-axis unidirectional composites. It holds promise for analysing multiaxial fatigue data. In fact, Chu et al. [16] have shown that the SWT parameter, in a different form, satisfactorily models the stress–strain behaviour and fatigue-life prediction of steel shafts subjected to multiaxial loading. In their case, since the material was isotropic, only the macrostresses and strains were considered. Typical of other interactive failure models for orthotropic unidirectional composites the present fatigue parameter does not distinguish between failure mechanisms for a given fibre/load angle. However, since matrix failure is dominant, comparison may be made with simple interactive models for static loading such as the Pucksimple failure criterion [17] derived on the assumption that matrix failures are determined by s22 and t12, and fibre failures are determined by s11:
冉 冊 冉冊
s22 2 t12 2 ⫹ ⫽1 Y Q
(14)
where s22 and Y are the respective tensile and failure stresses normal to the fibre (in-plane), and t12 and Q are the respective shear and failure stresses in the 1–2 plane. Eq. (14) can be rearranged assuming linear elastic behaviour
s22e22⫹t12g12K⫽K⬘
(15)
where ⑀22 and E22 are the strain and elastic modulus normal to the fibre (in-plane), g12 and G12 are the shear strain and modulus in the 1–2 plane, K=G12 Y2/E22 Q2 and K⬘=Y2/E22. Eqs. (6) and (15) have the same form and respectively express the combination of the tensile and shear stresses and strains required to initiate failure under static and cyclic loading conditions in the matrix of an off-axis unidirectional composite. 5. Conclusions A parameter has been proposed to predict the fatigue lives of unidirectional composites loaded off-axis. The fatigue parameter includes the maximum shear and normal stresses as well as the normal and shear stress ranges on the fracture plane parallel to the fibre axis. It is load path dependent. This approach could possibly reduce the number of fatigue tests required to characterize a particular fibre/load orientation or mean stress for an off-axis unidirectional composite. The fatigue parameter has been applied to predict the lives of unidirectional E-glass epoxy reinforced composites cycled at different fibre/load angles and mean
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Fig. 5.
855
Unified fatigue parameter (⌬W*) plotted against number of reversals to failure (2Nf) for different stress ratios.
conditions it holds promise for application to those cases where unidirectional composites are used for components that are cycled under multiaxial loading conditions.
Acknowledgements
Fig. 6. Predicted curve for Smax versus 2Nf for R=0.5 and q=71° data using R=0 and q=45° data.
stresses. The predicted data are in good agreement with the experimental results. It is shown that by applying the fatigue parameter derived from experimental data for one fibre/load angle, the fatigue lives of the same composite with other fibre/load angles may be predicted. A similar approach may be applied for mean stresses. Since the fatigue parameter appears capable of correlating fatigue data obtained under a variety of loading
This study was supported by the National Science and Engineering Research Council of Canada (NSERC) under Grant 0GP 0002770 and National Nature Science Foundation of China (NNSFC). The authors wish to thank Dr G. Wang, Dr L. Shi, Mr Y. Huang and Mr J. Petermann for valuable discussions and Marlene Dolson for typing the manuscript.
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